Problem: Find the least common multiple $(\text{LCM})$ of $3z^3-6z^2-9z$ and $7z^4+21z^3+14z^2$. You can give your answer in its factored form.
Explanation: The least common multiple $(\text{LCM})$ of two polynomial expressions is the polynomial with the least number of factors that is divisible by both polynomials. [How does this relate to the least common multiple of integers?] We can find the $\text{LCM}$ by factoring the two polynomials as much as possible and then comparing the factors: $3z^3-6z^2-9z$ can be factored as ${(3)(z)}{(z+1)}{(z-3)}$ by factoring out a $3z$ and using the sum-product pattern. $7z^4+21z^3+14z^2$ can be factored as ${(7)(z^2)}{(z+1)}{(z+2)}$ by factoring out a $7z^2$ and using the sum-product pattern. We can see that: Both polynomials share the factor ${(z+1)}$ Both polynomials share the factor ${(z)}$ Only the first polynomial has the factors ${(3)(z-3)}$ Only the second polynomial has the factors ${(7)(z+2)}$ and another factor of ${(z)}$ Therefore, the least common multiple is the product of all the above factors: [Why?] $\begin{aligned}&\phantom{=}{(z+1)(z)}{(3)(z-3)}{(7)(z)(z+2)}\\\\ &=21z^2(z+1)(z+2)(z-3)\end{aligned}$ In conclusion, the least common multiple of the two polynomials is $21(z^2)(z+1)(z+2)(z-3)$.